Algebraic spectral synthesis and crystal rigidity
type: Item
instance of: scholarly article
title: Algebraic spectral synthesis and crystal rigidity
zbMATH Open document ID: 1428.43006
author: Eleftherios Kastis
author: S. C. Power
published in: Journal of Pure and Applied Algebra
publication date: 2019-07-03T00:00:00Z
review text: A theorem of \textit{M. Lefranc} [C. R. Acad. Sci., Paris 246, 1951--1953 (1958; Zbl 0084.10302)] states that in the topology of pointwise convergence, every closed translation invariant linear space of complex-valued functions on \(\mathbb{Z}^d\) is the closed linear span of a set of exponential monomials \[ \mathbb{Z}^d \leftrightarrow \mathbb{C} , \quad (k_1,\ldots , k_d)\mapsto h(k_1, \ldots , k_d)\omega_1^{k_1}\cdots \omega_d^{k_d}, \] where \(h(k_1, \ldots , k_d)\) is a polynomial in \(d\) variables and \(\omega_i\), \(i=1, \ldots , d\), are nonzero complex numbers. In paper under review, the authors give a generalization of Lefranc's result for the closed invariant subspaces of \(C(\mathbb{Z}^d , \mathbb{C}^r)\), the space of vector-valued functions \(u \colon \mathbb{Z}^d \leftrightarrow \mathbb{C}^r\) with the topology of coordinatewise convergence. More precisely, let \(A\) be a closed invariant subspace of \(C(\mathbb{Z}^d, \mathbb{C}^r)\). Then there is a finite set of geometric indices such that \(A\) is the closed linear span of vectorial \(pg\)-sequences in \(A\) with geometric indices in this set (Theorem 3.10). In the last section of the paper the authors discuss connections of their results to difference equations and crystal flexibility. The paper contains a brief historical discussion of Lefranc's result.
reviewed by: M. Sangani Monfared
Mathematics Subject Classification ID: 43A45
Mathematics Subject Classification ID: 13E05
Mathematics Subject Classification ID: 52C25
zbMATH DE Number: 7076026
zbMATH Keywords: spectral synthesis
zbMATH Keywords: discrete group
zbMATH Keywords: shift-invariant subspace
zbMATH Keywords: crystal rigidity
MaRDI profile type: Publication
full work available at URL: j.jpaa.2019.03.003
OpenAlex ID: W2922346495
cites work: Q3064461
cites work: Q5563439
cites work: The almost periodic rigidity of crystallographic bar-joint frameworks
cites work: Q5465277
cites work: Fundamental Solutions for Multivariate Difference Equations
cites work: Q4731622
cites work: Periodic frameworks and flexibility
cites work: Q3999065
cites work: Q5512374
cites work: Q5512375
cites work: L'algèbre de Fourier d'un groupe localement compact
cites work: Spectral Synthesis Problems for Invariant Subspaces on Groups. I .
cites work: Q4279195
cites work: Infinitesimal rigidity for non-Euclidean bar-joint frameworks
cites work: Ideals and differential operators in the ring of polynomials of infinitely many variables
cites work: Spectral synthesis on discrete Abelian groups
cites work: Q3254054
cites work: Q5660121
cites work: Infinite bar-joint frameworks, crystals and operator theory
cites work: Crystal frameworks, symmetry and affinely periodic flexes
cites work: Introduction to shift-invariant spaces. Linear independence.
cites work: Q3407081
DOI: 10.1016/J.JPAA.2019.03.003
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description: scientific article; zbMATH DE number 7076026